Weak Landau-Ginzburg models for smooth Fano threefolds

摘要

The paper is joined with arXiv:0911.5428 and improved. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds withPicard rank 1 can be represented as Laurent polynomials in 3 variablesexhibiting them case by case. We check that these Landau-Ginzburg models can becompactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov'sprogram we prove that numbers of irreducible components of the central fibersof compactifications of these pencils are dimensions of intermediate Jacobiansof Fano varieties plus 1. In particular these numbers do not depend oncompactifications. We state most of known methods of finding Landau-Ginzburgmodels in terms of Laurent polynomials. We discuss Laurent polynomialrepresentation of Landau-Ginzburg models of Fano varieties and state someproblems related to it.